Can any one help me with my math Homework ?
By rithuvysakh
@rithuvysakh (25)
India
February 12, 2011 2:19am CST
Here's the question :- If a,b,c are positive and non-zero integers, then prove that
the value of the determinant
|a b c|
|b c d| is negative ?
|c a b|
1 response
@owlwings (43910)
• Cambridge, England
12 Feb 11
These sites may help:
http://www.college-cram.com/study/algebra/matrix-algebra/determinant-of-a-3x3-matrix/
http://en.wikipedia.org/wiki/Determinant
You need to prove that:
The expression (ada + bbb + ccc) is always greater than (acb + bdc + cba) for any positive and non-zero values of a, b and c.
@owlwings (43910)
• Cambridge, England
12 Feb 11
If the second line of the matrix should have been 'bca', then (a^3 + b^3 + c^3) should be greater than (3abc). That should not be too difficult to prove.
@owlwings (43910)
• Cambridge, England
12 Feb 11
If a = b = c = 1 (the minimum that meets the criteria for all variables), then both expressions equal 3 and the determinant is zero (which can, of course, be negative).
If any one of the variables is greater than 1, (a^3 + b^3 + c^3) must always be greater than (3abc) for the statement to be true.
I can easily see that, empirically, that is true - finding an absolute proof may be more difficult.
